Question

What is the domain and range of `y= ((x+1)(x-5)) /( x(x-5)(x+3))`?

Answer 1

Since this is a rational function, the domain will include undefined points on the graph called asymptotes.

Explanation:

Vertical asymptotes

Vertical asymptotes occur when the denominator is 0. Often, you will need to factor the denominator, but this has already been done.

`x(x - 5)(x + 3) -> x != 0, 5, -3`

Thus, you have your vertical asymptotes.

Your domain will be `x != 0, x!=5, x!=-3`

Horizontal Asymptotes:

The horizontal asymptotes of a rational function are obtained by comparing the degrees of the numerator and the denominator.

Multiplying everything out of factored form, we find that the degree of the numerator is 2 and that of the denominator is 3.

In a rational function of the form `y = (f(x))/(g(x))`, if the degree of `f(x)` is larger than that of `g(x)`, there will be no asymptote. If the degrees are equal, then the horizontal asymptote occurs at the ratio of the coefficients of the highest degree terms. If the degree of g(x) is smaller than `f(x)` there is an asymptote at y = 0.

Picking which scenario applies for our function, we realise there will be a vertical asymptote at `y =0`

Thus, our range is `y !=0`

Hopefully this helps!